Expert: Scotto - 8/5/2009

Question
a candy box is made from a piece of cardboard that measures 13 by 8 inches.  Squares of equal size will be cut out of each corner.  the sides will then be folded up to form a rectangle box.  What size should be cut from each corner to obtain max volume?

Answer
The box is 13 by 8, and the squares will be have a length of a side known as x.
Since for each side there is a square at one end and a square at the other,
that makes each side -2x, so the sides have length 5-2x and 8-2x.

The total volume of such a box would be x(5-2x)(8-2x).
It can be seen that the volume is 0 if x = 0 or x = 5/2.
If x = 0, the function is multiplied by x, so the function is 0.
If x = 5/2, the function is multiplied by (5-2x), which is 5 - 5*2/2 = 5-5 = 0.

To guess at the maximum, average these two points together and the answer is 5/4.
That gives us a ballpark to shoot for.  The actual value of x should be somewhat less than this since it is the height, but is subtracted from both sides.

If the last two terms of the function are multiplied out, (5-2x)(8-2x) = 40 - 26x + 4x².
Multiplying this by x gives 40x - 26x² + 4x³.

We'll call this f(x).  So we have f(x) = 40x - 26x² + 4x³.
Now f'(x) = 40 - 52x + 12x².  At first I see that all the terms are divisible by 4.
That gives f'(x) = 4(10 - 13x + 3x²).  OK, so I'll fall back on the quadratic equation.
You know, x = (-b ± √(b² - 4ac))/(2a) where a = 3, b = -13, and c = 10.
That gives me x = (13 ± √(169 - 120))6 = (13 ± √49)/6 = (13 ± 7)/6 = 6/6 and 20/6,
or x = 1 and 10/3.

NOw since 10/3 is 3.3333... and the number we're looking for needs to be between 1 and 5/2, we can throw out that one.  Doing so shows us the only answer that makes sense is 1.

Here we have a list of the side lengths with the resulting volume.
0.0   8.0   5.0   0
0.1   7.8   4.8   3.744
0.2   7.6   4.6   6.992
0.3   7.4   4.4   9.768
0.4   7.2   4.2   12.096
0.5   7.0   4   14
0.6   6.8   3.8   15.504
0.7   6.6   3.6   16.632
0.8   6.4   3.4   17.408
0.9   6.2   3.2   17.856
1.0   6.0   3   18
1.1   5.8   2.8   17.864
1.2   5.6   2.6   17.472
1.3   5.4   2.4   16.848
1.4   5.2   2.2   16.016
1.5   5.0   2   15
1.6   4.8   1.8   13.824
1.7   4.6   1.6   12.512
1.8   4.4   1.4   11.088
1.9   4.2   1.2   9.576
2.0   4.0   1   8
2.1   3.8   0.8   6.384
2.2   3.6   0.6   4.752
2.3   3.4   0.4   3.128
2.4   3.2   0.2   1.536

As can be seen, x=1 gives us the largest volume.